This lemma is a variant of the so-called stress-energy identity,
see e.g. [1]} and
[2]}.
To simplify the notation, we write \(D_j := D_{\tau _j}\) .
For a scalar function \(\varphi \) , we write
\(\nabla _j := (\nabla \varphi , \, \tau _j)_g = \mathrm {d}\varphi (\tau _j)\) .
Under the assumption \(f_\varepsilon \in L^2_{\operatorname{tan}}(M)\) , the second
derivatives of \(u_\varepsilon \)  exist in \(L^2_{\operatorname{tan}}(M)\) , by elliptic regularity.
Therefore, we can compute the gradient of the energy
density \(e_\varepsilon (u_\varepsilon )\) .
We have
\(\begin{split}\nabla _j \left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g\right)&= (D_jD_k u_\varepsilon , \, D_k u_\varepsilon )_g= \nabla _k (D_j u_\varepsilon , \, D_k u_\varepsilon )_g - (D_j u_\varepsilon , \, D_kD_k u_\varepsilon )_g \\&\hspace{-5.69046pt}\stackrel{(\ref {eq:rough_frame})}{=}\nabla _k (D_j u_\varepsilon , \, D_k u_\varepsilon )_g - (D_j u_\varepsilon , \, \Delta _g u_\varepsilon )_g+ (\mathrm {div}\,\tau _k) \, (D_j u_\varepsilon , \, D_k u_\varepsilon )_g\end{split}\) 
The equation (REF ) implies
\(\begin{split}\nabla _j \left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g\right)= \nabla _k (D_j u_\varepsilon , \, D_k u_\varepsilon )_g &+ (D_j u_\varepsilon , \, f_\varepsilon )_g- \frac{1}{\varepsilon ^2}(\left| u_\varepsilon  \right|^2 - 1) \, (D_j u_\varepsilon , \, u_\varepsilon ) \\&+ (\mathrm {div}\,\tau _k) \, (D_j u_\varepsilon , \, D_k u_\varepsilon )_g\end{split}\) 
and hence,
\( \begin{split}\nabla _j e_\varepsilon (u_\varepsilon )= \nabla _k (D_j u_\varepsilon , \, D_k u_\varepsilon )_g + (D_j u_\varepsilon , \, f_\varepsilon )_g+ (\mathrm {div}\,\tau _k) \, (D_j u_\varepsilon , \, D_k u_\varepsilon )_g\end{split}\) 
We integrate by parts, as in (REF ) and (REF ):
\(\begin{split}&\int _U\left(e_\varepsilon (u_\varepsilon ) \, (\mathrm {div}\,X) - (D_{j}u_\varepsilon , \, D_{k}u_\varepsilon )_g\, (\tau _j, \, D_{k} X)_g \right) \mathrm {vol}_g\\&\quad = \int _U \left( - \nabla _j e_\varepsilon (u_\varepsilon )\, (X, \, \tau _j)_g+ \nabla _k (D_{j}u_\varepsilon , \, D_{k}u_\varepsilon )_g \, (\tau _j, \, X)_g+ (D_{j}u_\varepsilon , \, D_{k}u_\varepsilon )_g \, (D_k\tau _j, \, X)_g \right) \mathrm {vol}_g\\&\quad \quad + \int _U (\mathrm {div}\,\tau _k) \, (D_{j}u_\varepsilon , \, D_{k}u_\varepsilon )_g \, (\tau _j, \, X)_g \, \mathrm {vol}_g\\&\hspace{85.35826pt} + \int _{\partial U} \left(e_\varepsilon (u_\varepsilon ) \, (X, \, \nu )_g - (D_{j}u_\varepsilon , \, D_{k}u_\varepsilon )_g\, (\tau _j, \, X)_g \, (\tau _k, \, \nu )_g \right) \mathrm {d}{H}^1\end{split}\) 
By substituting (REF ) in the right-hand side,
the lemma follows.
